Accuracy analysis of the percentile method for estimating non normal manufacturing quality

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(1)This article was downloaded by: [National Chiao Tung University 國立交通大學] On: 26 April 2014, At: 00:39 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK. Communications in Statistics - Simulation and Computation Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lssp20. Accuracy Analysis of the Percentile Method for Estimating Non Normal Manufacturing Quality a. b. c. Chien-Wei Wu , W. L. Pearn , C. S. Chang & H. C. Chen. b. a. Department of Industrial Engineering & Systems Management , Feng Chia University , Taiwan b. Department of Industrial Engineering & Management , National Chiao Tung University , Taiwan c. Department of Industrial Engineering & Management , Cheng Kuo Institute of Technology , Taiwan Published online: 07 May 2007.. To cite this article: Chien-Wei Wu , W. L. Pearn , C. S. Chang & H. C. Chen (2007) Accuracy Analysis of the Percentile Method for Estimating Non Normal Manufacturing Quality, Communications in Statistics - Simulation and Computation, 36:3, 657-697, DOI: 10.1080/03610910701212785 To link to this article: http://dx.doi.org/10.1080/03610910701212785. PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &.

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(3) Communications in Statistics—Simulation and Computation® , 36: 657–697, 2007 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0918 print/1532-4141 online DOI: 10.1080/03610910701212785. Quality Control. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Accuracy Analysis of the Percentile Method for Estimating Non Normal Manufacturing Quality CHIEN-WEI WU1 , W. L. PEARN2 , C. S. CHANG3 , AND H. C. CHEN2 1. Department of Industrial Engineering & Systems Management, Feng Chia University, Taiwan 2 Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan 3 Department of Industrial Engineering & Management, Cheng Kuo Institute of Technology, Taiwan Vännman (1995) proposed a superstructure Cp u v of capability indices for processes with normal distributions, which include Cp , Cpk , Cpm , and Cpmk as special cases. Pearn and Chen (1997) considered a generalization of Cp u v, called CNp u v, to cover processes with non normal distributions. Pearn and Chen (1997) also proposed a sample percentile estimator for the generalization CNp u v. In this article, we investigate the performance of the sample percentile estimator. We perform some simulation study, which covers the normal distribution and various non normal distributions including the uniform distribution, chi-square distribution, student’s t distributions, F distribution, beta distribution, gamma distribution, Weibull distribution, lognormal distribution, triangular distribution, and Laplace distribution, with selected parameter values. Extensive simulation results, comparisons, and analysis are provided. Keywords Flexible capability indices; Non normal distributions; Relative bias; Sample percentile estimator; Simulation. Mathematics Subject Classification Primary 62G05; Secondary 62P30.. 1. Introduction Process capability indices Cp u v, which include the two basic indices Cp and Cpk (Kane, 1986), and the two more-advanced indices Cpm and Cpmk (Chan et al., 1988; Pearn et al., 1992) as special cases, have been proposed in the Received July 6, 2006; Accepted November 13, 2006 Address correspondence to Chien-Wei Wu, Department of Industrial Engineering and Systems Management, Feng Chia University, 100, Wenhwa Road, Seatwen, Taichung 40724, Taiwan; E-mail: [email protected]. 657.

(4) 658. Wu et al.. manufacturing industry to provide numerical measures on process potential and process performance. The superstructure Cp u v has been defined as the following (see Vännman, 1995):. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. d − u − m Cp u v = 3 2 + v − T2 where is the process mean, is the process standard deviation, d = USL − LSL/2 is half length of the specification interval LSL USL, m = USL + LSL/2 is the mid-point between the upper and the lower specification limits, T is the target value, and u v ≥ 0. By setting u v = 0 and 1, we obtain the four indices, Cp 0 0 = Cp , Cp 1 0 = Cpk , Cp 0 1 = Cpm , and Cp 1 1 = Cpmk . These four indices, have been investigated extensively by Kane (1986), Chan et al. (1988), Pearn et al. (1992, 1998), Kotz et al. (1993), and many others. For thorough discussions of different capability indices and their statistical properties, see, e.g., the books by Kotz and Johnson (1993) and Kotz and Lovelace (1998), and the review article with discussion by Kotz and Johnson (2002). The index Cp only considers the process variability , thus provides no sensitivity on process departure from the target value at all. The index Cpk takes the process mean into consideration but it can fail to distinguish between on-target processes from off-target processes. The index Cpm takes proximity of process mean from the target value into account, and is more sensitive to process departure than Cp and Cpk . The index Cpmk adds an additional term − T2 in the definition, as a penalty to the process quality due to the departure of process mean from the target value. This additional penalty will be more sensitive to departure and therefore is able to distinguish better between off-target and on-target processes. The formulae for these indices are easy to understand and straightforward to apply. In practice, sample data must be collected to calculate these indices since the process mean and process variance 2 are usually unknown, while the X 2 /n − 1, sample mean X = ni=1 Xi /n, and the sample variance S 2 = ni=1 Xi − are the conventional estimators of process mean and process variance 2 . For normal distributions, these estimators based on X and S 2 are quite stable and reliable. However, for non normal distributions, they become highly unstable since the distribution of the sample variance, S 2 , is sensitive to departures from normality. Somerville and Montgomery (1996–97) presented an extensive study to illustrate how poorly the normally based capability indices perform as a predictor of process fallout when the process is non normally distributed. If the normally based capability indices are still used to deal with non normal process data, the values of the capability indices are incorrect and might misrepresent the actual product quality. Clements (1989) proposed a conceptually simple method for calculating Cp and Cpk , for any shape of distribution, using the Pearson family of curves. Pearn and Kotz (1994–95) adopted Clements’ method to modify normality-based capability indices into non normality Pearson-based capability indices for the Cpm and Cpmk indices. The advantage of Clements’ method is that it is easy to understand and apply. But somewhat limited as not all data distributions can be described adequately with a Pearson or Johnson curve. Chang and Lu (1994) also applied Clements’ approach and proposed a percentile method for calculating PCIs without assuming normal distribution. Moreover, Pearn and Chen (1997) extended their.

(5) Accuracy Analysis of the Percentile Method. 659. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. method to the generalization CNp u v and constructed a superstructure for the CNp u v. Although they proposed these estimators, estimators of CNp u v, namely, they did not show how good it is, i.e., relative bias. In this article, we first give a brief introduction to the flexible capability index and then introduce their estimators based on the percentile method in Sec. 3. Subsequently, in Sec. 4, an extensive simulation study was conducted to examine the performance of the estimated CNp u v for 11 distributions (which cover processes with normal and various non normal distributions). In Sec. 5, the performance analysis of the percentile estimators P0135 , M, P99865 and the estimators of Cp u v are presented based on the relative bias. In Sec. 6, an illustrative example is presented for demonstrating how we apply the percentile method to the actual data taken from a factory. Finally, some concluding remarks are made in Sec. 7.. 2. Flexible Capability Index CNp u v The indices Cp u v are appropriate for processes with normal distributions, but have been shown to be inappropriate for processes with non normal distributions. Pearn and Chen (1997) considered the following generalization of Cp u v, called CNp u v, to cover processes with non normal distributions. d − uM − m CNp u v = 2 3 P998656−P0135 + vM − T2 where P is the -percentile, M is the median of the distribution, m and T are defined as before. In developing the generalization CNp u v, Pearn and Chen (1997) replaced the process mean by the process median M (a more robust measure for process central tendency), and the process standard deviation by P99865 − P0135 /6 in the definition of the original indices Cp u v. We note that P99865 − P0135 covers a probability of 99.73% for any distributions. In the special case where the underlying distribution is normal, then M = and P99865 − P0135 = 6. Clearly, the generalization of CNp u v will reduce to Cp u v. By setting u v = 0 and 1, we obtain CNp 0 0 = CNp , CNp 1 0 = CNpk , CNpm 0 1 = CNpm , and CNp 1 1 = CNpmk , which can be expressed as the following: USL − LSL P99865 − P0135      USL − M M − LSL  = min    3 P99865 −P0135 2 3 P99865 −P0135 2  6 6. CNp =. CNpk. USL − LSL CNpm = 2 P99865 −P0135 2 M + − T 6 6 USL − M M − LSL CNpmk = min 2 2 P99865 −P0135 2 P99865 −P0135 2 3 + M − T 3 + M − T 6 6 Pearn and Chen (1997) showed that the generalization CNp u v is more consistent and more accurate than the original indices Cp u v in measuring process.

(6) 660. Wu et al.. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Figure 1. Distribution plots of processes A, B, and C.. capability for non normal distributions. To compare the original indices Cp u v and the generalization CNp u v, we consider an example of three processes A, B, and C, depicted in Fig. 1. All three processes are distributed as chi-square distributions with 3 degrees of freedom (heavily skewed with long tails). While process B is on-target B = T, processes A and C are severely off-target (A = LSL and C = USL. Table 1 displays the characteristics of processes A, B, and C, including the process mean, standard deviation, median and percentiles. The proportion of non-conformity for process A is 61%, which is significantly greater than that, 39%, for process C. But, both processes A and C obtain the same original index values, thus the original index Cp u v inconsistently measures the process capability. On the other hand, the generalization CNp u v clearly differentiates A and C by giving small value to A and larger value to C. Table 2 summarizes the value of two generalize indices, Cp u v and CNp u v, for the three processes A, B, and C. Furthermore, for processes distributed as Weibull, the result is also the same. In fact, for Weibull distributions W with = 3 and = 11, the percentage comparisons, 61% versus 39%, displayed in Fig. 1 will be replaced by 62% versus 38%.. Table 1 Characteristics of processes A, B, C Process A B C. . . P0135. M. P99865. 1000 1800 2600. 245 245 245. 773 1573 2300. 937 1737 2537. 2263 3063 3463. Table 2 A comparison between Cp u v and CNp u v for processes A, B, C Process A B C. Cp. Cpk. Cpm. Cpmk. CNp. CNpk. CNpm. CNpmk. 1.09 1.09 1.09. 0.00 1.09 0.00. 0.27 1.09 0.27. 0.00 1.09 0.00. 1.07 1.07 1.07. −008 099 008. 0.30 1.04 0.34. −002 096 003.

(7) Accuracy Analysis of the Percentile Method. 661. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 3. Percentile Estimator of CNp u v Pearn and Chen (1997) considered a sample percentile estimator to calculate the index CNp u v. The estimator essentially applies the sample percentile method proposed by Chang and Lu (1994) for calculating P99865 , P0135 , and the median M. The estimator can be expressed as: d − u M − m CNp u v = 2 2 P99865 − P0135 3 +v M −T 6 99865n + 0135 P99865 = XR1 + − R1 × XR1 +1 − XR1 100 0135n + 99865 P0135 = XR2 + − R2 × XR2 +1 − XR2 100 n+1 M = XR3 + − R3 × XR3 +1 − XR3 2 99865n + 0135 R1 = 100 0135n + 99865 n+1 R2 = R3 = 100 2 In this setting, the notation R is defined as the greatest integer less than or equal to the number R, and Xi is defined as the ith order statistic. The sample percentile estimator is easy to understand and straightforward to apply. Since the calculation does not require any knowledge of the process distribution, then the estimator may be used on any process regardless of whether the underlying distribution is normal or non normal. Several other alternatives for estimating the percentile points and those indices have been proposed, which are described below. Those alternatives are either inefficient, or require a large amount of data. Therefore, those alternatives are inferior to the percentile estimator (which is applicable for any sample size), and are not appropriate for factory applications. Clements Method. If the underlying process distribution is of Pearson type, Clements (1989) proposed a method for estimating capability indices, CNp and CNpk . The method first utilizes estimates of the mean ( X , standard deviation (S, skewness (sk), and kurtosis (ku), to determine the type of the Pearson distribution curve. The method then utilizes the tables provided by Gruska et al. (1989) for percentages of the family of Pearson curves as a function of skewness and kurtosis. Pearn and Kotz (1994–95) extended the method to the other two indices, CNpm and CNpmk . The proposed method, however, requires that the underlying process distribution be Pearson type, and a large number of tables provided by Gruska et al. (1989). Zwick Method. For non normal process distributions, Zwick (1995) proposed a method to fit continuous probability distributions for given data. The method essentially calculates the uniform order statistic medians for the inverse or the percentile function. Traditional method, the maximal likelihood estimate (MLE), modified moment estimator (MME) are used for estimating the parameters..

(8) 662. Wu et al.. are then calculated The percentile points P99865 , P0135 , and the sample median M, from the fitted continuous probability distribution function.. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Schneider et al. Method. Schneider et al. (1995) pointed out that percentiles are not statistically efficient estimates, and proposed to take a sample of size n ≥ 1/000135 then use maximum and minimum data values for the percentiles P99865 and P0135 . This method is obviously not efficient as it requires a large amount of data, although may be efficient in supplier certification process where enough data are usually available. Sarkar and Pal Method. If the underlying process distribution is of Extreme Value distribution type whose parameter and can be estimated from the sample, Sarkar and Pal (1997) proposed a method for estimating the one where can sided capability index CPU = USL − M/ P99865 − M, P99865 and M = be calculated as the following: P99865 = − ln−ln099865 = + 03665 , M − ln−ln05 = + 6607 . If the underlying process distribution is not of extreme value distribution type, the calculation can be misleading.. 4. Empirical Distribution of Percentile Estimator The exact distribution of the percentile estimator, CNp u v, is mathematically intractable even under normality assumption. We therefore use simulation technique to investigate the accuracy of the estimator CNp u v based on the relative bias for a wide variety of distributions. The relative bias was defined as the bias of CNp u v − the estimated E CNp u v divided by the true value of CNp u v,

(9) E CNp u v /CNp u v. It cannot only provide a measure of the magnitude of the bias but also facilitate interpretation of the bias on an appropriate scale. The distributions in our investigation including: (1) the normal distribution N 2 ; (2) the uniform distribution Ua b; (3) the Laplace distribution La b; (4) the Student’s t distribution tr; (5) the chi-square distribution 2 r; (6) the F distribution Fr1 r2 ; (7) the beta distribution B ; (8) the gamma distribution G ; (9) the Weibull distribution W ; (10) the lognormal distribution LN 2 ; and (11) the Triangular distribution Ta b c, with some selected parameter values. (1) Normal distribution, N 2 , with probability density function fx = 21/2 −1 exp−x − 2 /22 for − < x < , > 0. Parameters are set to = 0, and = 1. (2) Uniform distribution, Ua b, with probability density function fx = 1/b − a, for a ≤ x ≤ b, − < a < b < . Parameters are set to a = 0, b = 1. (3) Laplace distribution, La b, with probability density function fx = exp−x − a/b/2b for all x. Parameters are set to a = 0, b = 1. (4) Student’s t distribution, tr, with probability density function fx = r + 1/2r1/2 r/2−1 1 + x2 /r−r+1/2 for − < x < . Degrees of freedom are set to r = 4, 5, 6, 7, 8. (5) Chi-square distribution, 2 r, with probability density function fx = 2−r/2 xr/2−1 exp−x/2/r/2 for 0 < x < . Degrees of freedom are set to r = 3 4 5 6 7..

(10) Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Accuracy Analysis of the Percentile Method. 663. (6) F distribution Fr1 r2 , with probability density function fx = r1 + r2 /2r1 /r2 r1 /2 xr1 /2−1

(11) r1 /2r2 /21 + r1 /r2 xr1 +r2 /2 −1 , for x > 0. Degree of freedoms are set to r1 = 10, and r2 = 10 20 30 40 50. (7) Beta distribution B , with probability density function fx = x −1 1 − x −1 + / for 0 ≤ x ≤ 1. Parameters are set to = 3,. = 2.0, 2.5, 3, 3.5, 4.0. (8) Gamma distribution G , with probability density function fx =. − x −1 e−x/ / for x > 0. Parameters are set to = 15 20 25 30 35 and = 1. (9) Weibull distribution W , with probability density function fx =. x −1 exp− x for x ≥ 0. Parameters are set to = 1, = 14 16 18 20 22. (10) Lognomal distribution LN 2 , with probability density function fx = exp−ln x − 2 /22 /21/2 x for x ≥ 0. Parameters are set to = 0, 2 = l, 1/4, 1/9, 1/16, 1/25. (11) Triangular distribution Ta b c, with probability density function fx = 2x − a/b − ac − a for a ≤ x ≤ c, fx = 2b − x/b − ab − c for c < x ≤ b, a < c < b. Parameters are set to a = 0, b = 1, c = 1/4 1/3 1/2 3/4 5/6. Figures 2–9 display the probability density function (pdf) plots for simulated distributions. Figure 2 plots the probability density function for standard normal, uniform U0 1, Laplace L0 1, and Student’s t4 distributions. Figure 3 plots the chi-square distributions 2 r with selected degrees of freedom = 3 4 5 6 7 (from left to right in plot). Figure 4 plots the F distributions Fr1 r2 with selected degrees of freedom parameters 1 = 10 and 2 = 10 20 30 40 50 (from bottom to top in plot). Figure 5 plots the beta distributions B with parameters = 3 and = 20 25 30 35 40 (from left to right in plot). Figure 6 plots the gamma distributions G with parameter = 15, 2.0, 2.5, 3.0, 3.5, and = 1 (from left to right in plot). Figure 7 plots the Weibull distributions W with parameters = 1 and = 14 16 18 20 22 (from left to right in plot). Figure 8 plots the lognormal distributions LN 2 for parameters = 0, and 2 = 1, 1/4, 1/9, 1/16,. Figure 2. The pdf of normal, uniform, Laplace, and t distributions..

(12) Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 664. Wu et al.. Figure 3. The pdf of 2 r with r = 3 4 5 6 7 (from left to right).. Figure 4. The pdf of Fr1 r2 with r1 = 10, and r2 = 10 20 30 40 50 (from bottom to top).. Figure 5. The pdf of B with = 3, = 20, 2.5, 3, 3.5, 4.0 (from left to right)..

(13) Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Accuracy Analysis of the Percentile Method. 665. Figure 6. The pdf of G with = 1.5, 2.0, 2.5, 3.0, 3.5, and = 1 (from left to right).. Figure 7. The pdf of W with = 1, = 14 16 18 20 22 (from left to right).. Figure 8. The pdf of LN 2 with = 0, 2 = l 1/4 1/9 1/16 1/25 (from bottom to top)..

(14) Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 666. Wu et al.. Figure 9. The pdf of Ta b c with a = 0, b = 1, c = 1/4, 1/3, 1/2, 3/4, 5/6 (from left to right).. 1/25 (from bottom to top in plot). Figure 9 plots the triangular distributions Ta b c for parameters a = 0, b = 1 and c = 1/4 1/3 1/2 3/4 5/6 (from left to right in plot). We have chosen those distributions with selected parameter values to represent slight, moderate, and severe departures from normality, as those distributions are known to have significantly different tail behaviors, which greatly influence the process capability calculations. The simulation is carried out using 15,000,000 random numbers generated from the uniform distribution U0 1, applying AS183 generator (Wichmann and Hill, 1987) with multiple seeds using IBM RISC/6000 workstation.. 5. Accuracy Analysis of Percentile Estimator Parameter values used in our simulation study have been described earlier. We first perform the simulation for the three percentile estimators, P0135 , M, P99865 , to investigate the accuracy and calculate the relative bias. Next, an extensive simulation study was conducted to examine the performance of the estimated CNp u v for the 11 distributions (covering normal distribution and various non normal distributions), in terms of the relative bias,

(15) E CNp u v − CNp u v /CNp u v, for various values of d/ and − T/. P99865 5.1. Accuracy Result for P0135 , M, For simplicity of the presentation, detailed results are presented here only for some selected parameters. Tables 3–11 display the performance results of the and three percentile estimators, P0135 , M, P99865 , from the simulation for all 11 distributions, in terms of the relative bias E P − P /P . The sample sizes in the simulation are set to n = 10, 20, 30, 50, 100, 300, 500, 1,000, and 3,000. It is clear that for the three percentile estimators P0135 M P99865 , the relative bias decreases as the sample size n increases. This can be understood since the three percentile estimators are consistent and asymptotically unbiased, hence become.

(16) Accuracy Analysis of the Percentile Method. 667. Table 3 and Relative bias of P0135 , M, P99865 , for normal, uniform, and Laplace distributions N0 1. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. n 10 20 30 40 50 100 200 300 500 1000 3000. U0 1. L0 1. (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) −489 −381 −324 −287 −259 −180 −114 −83 −56 −33 −13. ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗. −489 −381 −325 −287 −259 −180 −115 −84 −57 −34 −14. 67051 35107 23784 18004 14452 7282 3659 2435 1460 736 250. 00 00 00 00 00 00 00 00 00 00 00. −91 −48 −32 −24 −20 −10 −05 −03 −02 −01 00. −624 −513 −448 −403 −368 −262 −169 −123 −82 −50 −20. ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗. −624 −513 −448 −403 −368 −263 −169 −124 −84 −51 −21. more accurate as the sample size increases. If the relative bias is negative, it indicates that P underestimates the parameter. On the other hand, if the relative bias is positive it indicates that P overestimates the parameter. For the relative bias of marked “∗ ∗ ∗” in Tables 3–4, it indicates that the value does not exist because M M is 0. From Tables 3–11, we observe that the relative bias of the three percentile estimators all exceed 80% for sample size n no greater than 50, except for normal, t, chi-square, and the Laplace distributions. It is noted that the relative bias is greater than 100% for U (0, 1), 2 3, G(1.5, 1), and W (1, 1.4) when the sample size n is 300. Further, the relative bias of the three percentile estimators, are greater than 10% for U (0, 1), 2 3, G(1.5, 1), W (1, 1.4), W (1, 1.6), and W (1, 1.8) when the sample size n is 3,000. Thus, for non normal distributions, the three percentile estimators are indeed highly inaccurate. Np u v 5.2. Accuracy Result for C Tables 12(a)–22 display the results from the simulation for the relative bias of CNp u v, for all 11 distributions with various values of d/ = 2, 3, 4, 5, and − T/ = 0 05 10 15, and 2.0. Clearly, the relative bias decreases as the sample size n increases. It is noted that for the relative bias of CNp u v marked “∗ ∗ ∗” in Tables 12(a)–22, it indicates that the value is meaningless because CNp u v is 0. For instance, if d/ = 2 and − T/ = 2, then the value of relative bias is meaningless since CNpk = 0 and CNpmk = 0 for Student’s t, normal, uniform, Laplace distributions. For most cases, direction of the relative bias of CNp u v is positive except for the uniform, beta, and triangular distributions. For beta distribution CNpmk are B 30 30 and triangular distribution T 0 1 1/2, the bias of CNpk , and negative for large n when the process is on-target. For uniform distribution, U 0 1, CNpmk are negative when the process is on-target. Since CNpm and the bias of CNpk , the probability that the process is (or detected to be) on target is zero, the relative bias can be regarded as positive..

(17) 10 20 30 40 50 100 200 300 500 1000 3000. n. t(5). t(6). t(7). t(8). −673 −575 −513 −468 −131 −313 −196 −135 −86 −58 −25. ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗. −674 −575 −514 −468 −132 −314 −196 −135 −86 −59 −26. −638 −537 −476 −432 −397 −287 −181 −127 −82 −54 −23. ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗. −638 −537 −476 −432 −397 −287 −181 −128 −84 −55 −23. −614 −511 −450 −407 −373 −269 −171 −122 −80 −52 −22. ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗. −614 −511 −450 −407 −373 −269 −171 −122 −80 −52 −22. −595 −492 −431 −389 −356 −256 −163 −116 −76 −48 −20. ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗. −596 −492 −432 −389 −356 −256 −163 −117 −78 −49 −21. −582 −477 −418 −376 −343 −246 −157 −112 −74 −46 −19. ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗. −582 −478 −418 −376 −344 −246 −157 −113 −75 −47 −20. P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%). t4. Table 4 and P99865 , for t distribution, t with = 418 Relative bias of P0135 , M,. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 668 Wu et al..

(18) 10 20 30 40 50 100 200 300 500 1000 3000. n. 42. 52. 62. 72. 16581 9827 7224 5799 4874 2807 1576 1106 704 375 131. 4.5 2.2 1.5 1.1 0.9 0.4 0.2 0.1 0.1 0.0 0.0. −514 −420 −366 −329 −300 −214 −137 −100 −68 −41 −17. 7871 5032 3858 3186 2735 1669 979 700 455 251 92. 3.2 1.6 1.0 0.8 0.6 0.3 0.2 0.1 0.1 0.0 0.0. −482 −393 −342 −306 −279 −198 −127 −73 −63 −38 −16 4970 3312 2597 2177 1890 1189 713 514 338 189 71. 2.5 1.2 0.8 0.6 0.5 0.2 0.1 0.1 0.0 0.0 0.0. −457 −371 −323 −289 −263 −187 −120 −88 −59 −36 −15 3610 2469 1964 1661 1452 931 565 409 270 153 58. 2.0 1.0 0.7 0.5 0.4 0.2 0.1 0.1 0.0 0.0 0.0. −437 −354 −307 −275 −250 −178 −114 −83 −56 −34 −14. 2841 1978 1588 1352 1187 770 471 342 227 129 49. 1.7 0.8 0.6 0.4 0.3 0.2 0.1 0.1 0.0 0.0 0.0. −420 −340 −294 −263 −239 −170 −109 −80 −54 −33 −13. (%) (%) (%) (%) (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%). 12. Table 5 and Relative bias of P0135 , M, P99865 , for chi-square distribution, 2 with = 3 4 5 6, and 7. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Accuracy Analysis of the Percentile Method 669.

(19) 10 20 30 40 50 100 200 300 500 1000 3000. n. F10 20. F10 30. F10 40. F10 50. 2163 1493 1197 1019 894 580 353 254 166 92 31. 3.6 1.7 1.1 0.8 0.7 0.3 0.2 0.1 0.1 0.0 0.0. −625 −537 −480 −438 −404 −295 −187 −132 −86 −57 −24. 1978 1391 1124 962 849 555 342 248 165 94 36. 2.3 1.1 0.7 0.5 0.4 0.2 0.1 0.1 0.0 0.0 0.0. −519 −434 −383 −346 −318 −229 −147 −106 −71 −45 −19 1913 1354 1097 941 830 546 337 245 163 94 36. 1.9 0.9 0.6 0.5 0.4 0.2 0.1 0.1 0.0 0.0 0.0. −476 −394 −346 −312 −285 −205 −132 −96 −64 −40 −16 1877 1333 1082 928 819 539 333 242 161 92 35. 1.7 0.8 0.6 0.4 0.3 0.2 0.1 0.1 0.0 0.0 0.0. −454 −373 −327 −294 −268 −192 −123 −90 −60 −37 −15. 1856 1320 1073 921 813 536 331 241 160 92 35. 1.6 0.8 0.5 0.4 0.3 0.2 0.1 0.1 0.0 0.0 0.0. −440 −360 −315 −283 −258 −184 −118 −86 −58 −36 −15. (%) (%) (%) (%) (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%). F10 10. Table 6 and Relative bias of P0135 , M, P99865 , for F distribution, F10 2 with 2 101050. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 670 Wu et al..

(20) 10 20 30 40 50 100 200 300 500 1000 3000. n. B30 25. B30 30. B30 35. B30 40. 2971 2128 1727 1479 1303 852 524 381 253 144 55. −05 −02 −02 −01 −01 00 00 00 00 00 00. −111 −73 −57 −47 −41 −25 −15 −11 −07 −04 −01. 3054 2174 1759 1504 1324 863 529 385 256 145 55. −02 −01 −01 −01 00 00 00 00 00 00 00 -14.5 -10.0 -7.9 -6.7 -5.9 -3.7 -2.3 -1.6 -1.1 -0.6 -0.2. 3117 2209 1784 1523 1339 871 534 388 258 146 56. 00 00 00 00 00 00 00 00 00 00 00. −174 −123 −99 −85 −75 −49 −30 −22 −15 −08 −03 3168 2236 1803 1537 1351 878 537 390 259 147 56. 0.2 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0. −198 −143 −117 −100 −89 −59 −36 −27 −18 −10 −04. 3208 2258 1818 1549 1361 883 540 392 260 148 56. 0.3 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0. −218 −160 −132 −114 −101 −68 −42 −31 −21 −12 −05. (%) (%) (%) (%) (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%). B30 20. Table 7 and Relative bias of P0135 , M, P99865 , for beta distribution, B3 with = 200540. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Accuracy Analysis of the Percentile Method 671.

(21) 10 20 30 40 50 100 200 300 500 1000 3000. n. G20 1. G25 1. G30 1. G35 1. 16581 9827 7224 5799 4874 2807 1576 1105 704 376 133. 4.5 2.2 1.5 1.1 0.9 0.4 0.2 0.1 0.1 0.0 0.0. −514 −420 −366 −329 −300 −214 −137 −101 −68 −41 −17. 7871 5032 3858 3186 2735 1670 979 700 456 251 93. 3.2 1.6 1.0 0.8 0.6 0.3 0.2 0.1 0.1 0.0 0.0. −482 −393 −342 −306 −279 −199 −128 −93 −63 −38 −16 4970 3312 2597 2177 1890 1190 713 514 338 189 71. 2.5 1.2 0.8 0.6 0.5 0.2 0.1 0.1 0.0 0.0 0.0. −457 −371 −323 −289 −263 −187 −120 −88 −59 −36 −15 3610 2469 1964 1261 1452 931 565 409 270 153 58. 2.0 1.0 0.7 0.5 0.4 0.2 0.1 0.1 0.0 0.0 0.0. −437 −354 −307 −275 −250 −178 −114 −83 −56 −34 −14. 2841 1978 1588 1352 1187 770 471 342 227 129 49. 1.7 0.8 0.6 0.4 0.3 0.2 0.1 0.1 0.0 0.0 0.0. −420 −340 −294 −263 −239 −170 −109 −80 −54 −33 −13. (%) (%) (%) (%) (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%). G15 1. Table 8 and Relative bias of P0135 , M, P99865 , for gamma distribution, G 1 with = 150535. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 672 Wu et al..

(22) 10 20 30 40 50 100 200 300 500 1000 3000. n. W1 16. W1 18. W1 20. W1 22. 18935 11263 8267 6622 5554 3173 1768 1237 784 418 149. 3.2 1.5 1.0 0.8 0.6 0.3 0.2 0.1 0.1 0.0 0.0. −452 −363 −314 −280 −254 −180 −115 −84 −57 −34 −14. 12300 7695 5805 4735 4026 2387 1368 968 622 337 122. 2.2 1.1 0.7 0.5 0.4 0.2 0.1 0.1 0.0 0.0 0.0. −412 −328 −282 −251 −228 −160 −103 −75 −51 −31 −12 8766 5693 4383 3624 3112 1897 1109 791 514 282 103. 1.5 0.7 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.0 0.0. −378 −300 −257 −228 −207 −145 −93 −68 −46 −27 −11 6659 4451 3481 2908 2516 1566 930 668 437 242 89. 1.1 0.5 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.0. −350 −275 −236 −209 −189 −132 −84 −62 −42 −25 −10. 5302 3626 2871 2419 2106 1333 801 578 381 213 80. 0.7 0.4 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.0 0.0. −325 −255 −217 −192 −174 −121 −77 −57 −38 −23 −09. (%) (%) (%) (%) (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%) P0135 (%) M P99865 (%). W1 14. Table 9 and Relative bias of P0135 M, P99865 for Weibull distribution, W1 with = 140222. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Accuracy Analysis of the Percentile Method 673.

(23) 10 20 30 40 50 100 200 300 500 1000 3000. n. LN0 1/4. LN0 1/9. LN0 1/16. LN0 1/25. 4079 2546 1941 1600 1371 829 472 325 190 66 −31. 8.7 4.1 2.7 2.0 1.6 0.8 0.4 0.3 0.2 0.1 0.0. −722 −629 −567 −520 −481 −354 −225 −158 −102 −69 −30. 1171 831 676 581 513 339 210 153 101 59 23. 2.1 1.0 0.7 0.5 0.4 0.2 0.1 0.1 0.0 0.0 0.0. −498 −415 −365 −330 −302 −218 −140 −102 −68 −43 −18 661 486 402 349 311 209 131 95 64 37 14. 0.9 0.4 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.0. −375 −306 −267 −240 −219 −156 −100 −73 −49 −30 −12 458 342 285 249 222 151 95 69 46 27 11. 0.5 0.3 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0. −300 −242 −210 −188 −171 −121 −78 −57 −38 −23 −09. 350 264 221 193 173 118 74 54 36 21 08. 0.3 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0. −249 −200 −173 −154 −140 −99 −63 −46 −31 −19 −08. % % % % % P0135 % M P99865 % P0135 % M P99865 % P0135 % M P99865 % P0135 % M P99865 % P0135 % M P99865 %. LN0 1. Table 10 and Relative bias of P0135 M, P99865 for lognormal, LN0 2 with = 1 1/2 1/3 1/4, and 1/5. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 674 Wu et al..

(24) 10 20 30 40 50 100 200 300 500 1000 3000. n. T0 1 1/3. T0 1 1/2. T0 1 3/4. T0 1 5/6. 6410 4358 3431 2876 2494 1559 928 668 438 243 91. 1.9 1.0 0.7 0.5 0.4 0.2 0.1 0.1 0.0 0.0 0.0. −210 −143 −113 −95 −82 −52 −31 −22 −15 −08 −03. 6398 4358 3431 2876 2494 1559 928 668 438 243 91. 1.5 0.8 0.6 0.4 0.3 0.2 0.1 0.1 0.0 0.0 0.0. −198 −135 −106 −89 −77 −48 −29 −21 −14 −08 −03 6394 4357 3430 2876 2494 1559 928 668 438 243 91. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0. −171 −116 −92 −77 −67 −42 −25 −18 −12 −07 −02 6394 4357 3430 2876 2494 1559 928 668 438 243 91. −12 −06 −04 −03 −03 −01 −01 0.0 0.0 0.0 0.0. −120 −82 −64 −54 −47 −29 −18 −13 −08 −05 −02. 6394 4357 3430 2876 2494 1559 928 668 438 243 91. -1.3 -0.6 -0.4 -0.3 -0.3 -0.1 -0.1 0.0 0.0 0.0 0.0. −99 −67 −52 −38 −38 −24 −14 −10 −07 −04 −01. % % % % % P0135 % M P99865 % P0135 % M P99865 % P0135 % M P99865 % P0135 % M P99865 % P0135 % M P99865 %. T0 1 1/4. Table 11 and Relative bias of P0135 M, P99865 for triangular distribution, T0 1 c with c = 1/4 1/3 1/2 3/4, and 5/6. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. Accuracy Analysis of the Percentile Method 675.

(25) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 2573 2573 2573 2573 1632 1632 1632 1632 1267 1267 1267 1267 916 916 916 916 566 566 566 566 215 215 215 215 84 84 84 84 34 34 34 34. 0.0. 2573 2573 2573 2573 1632 1632 1632 1632 1267 1267 1267 1267 916 916 916 916 566 566 566 566 215 215 215 215 84 84 84 84 34 34 34 34. 0.5. 2573 2573 2573 2573 1632 1632 1632 1632 1267 1267 1267 1267 916 916 916 916 566 566 566 566 215 215 215 215 84 84 84 84 34 34 34 34. 1.0. CNp. 2573 2573 2573 2573 1632 1632 1632 1632 1267 1267 1267 1267 916 916 916 916 566 566 566 566 215 215 215 215 84 84 84 84 34 34 34 34. 1.5 2573 2573 2573 2573 1632 1632 1632 1632 1267 1267 1267 1267 916 916 916 916 566 566 566 566 215 215 215 215 84 84 84 84 34 34 34 34. 2.0 2714 2307 2373 2413 1417 1489 1525 1546 1115 1166 1191 1206 815 849 866 876 507 527 536 542 188 197 201 204 71 76 78 79 27 30 31 32. 0.0 2550 2559 2563 2565 1630 1631 1631 1631 1267 1277 1267 1267 916 916 916 916 566 566 566 566 215 215 215 215 84 84 84 84 35 35 34 34. 0.5 2573 2573 2573 2573 1632 1632 1632 1632 1268 1268 1267 1267 916 916 916 916 566 566 566 566 215 215 215 215 84 84 84 84 35 35 34 34. 1.0. CNpk. 2574 2573 2573 2573 1633 1632 1632 1632 1268 1268 1268 1267 917 916 916 916 566 566 566 566 215 215 215 215 85 84 84 84 35 35 35 34. 1.5. 0.0 2066 2066 2066 2066 1482 1482 1482 1482 1196 1196 1196 1196 888 888 888 888 558 558 558 558 213 213 213 213 84 84 84 84 34 34 34 34. 2.0 ∗∗∗ 2573 2573 2573 ∗∗∗ 1632 1632 1632 ∗∗∗ 1268 1268 1267 ∗∗∗ 916 916 916 ∗∗∗ 566 566 566 ∗∗∗ 215 215 215 ∗∗∗ 84 84 84 ∗∗∗ 35 35 34 1428 1428 1428 1428 1056 1056 1056 1056 876 876 876 876 677 677 677 677 447 447 447 447 180 180 180 180 72 72 72 72 30 30 30 30. 0.5 726 726 726 726 574 574 574 574 497 497 497 497 405 405 405 405 284 284 284 284 122 122 122 122 51 51 51 51 22 22 22 22. 1.0. CNpm. 400 400 400 400 331 331 331 331 294 294 294 294 206 206 206 206 178 178 178 178 79 79 79 79 34 34 34 34 14 14 14 14. 1.5 247 247 247 247 209 209 209 209 188 188 188 188 159 159 159 159 118 118 118 118 52 52 52 52 23 23 23 23 10 10 10 10. 2.0. 0.0 1763 1864 1915 1945 1291 1355 1386 1406 1052 1100 1124 1138 790 823 839 849 500 519 529 534 187 196 200 203 71 75 77 79 27 30 31 32. Table 12(a) CNpk CNpm , and CNpmk for Student’s t distribution, t4 Relative bias (%) of CNp . Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 1523 1485 1469 1460 1098 1081 1074 1070 900 890 886 884 688 684 682 681 450 449 448 448 180 180 180 180 73 73 72 72 30 30 30 30. 0.5 845 786 766 756 623 599 591 587 526 512 507 505 419 412 410 408 290 287 286 285 123 122 122 122 51 51 51 51 22 22 22 22. 1.0. CNpmk. 546 449 429 421 397 353 444 341 334 307 302 303 268 253 250 249 188 182 180 180 81 80 79 79 35 35 35 35 15 15 15 15. 1.5. ∗∗∗ 298 272 264 ∗∗∗ 233 221 217 ∗∗∗ 203 192 193 ∗∗∗ 168 164 162 ∗∗∗ 121 119 119 ∗∗∗ 53 53 53 ∗∗∗ 23 23 23 ∗∗∗ 10 10 10. 2.0. 676 Wu et al..

(26) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 1753 1753 1753 1753 1108 1108 1108 1108 862 862 862 862 626 626 626 626 392 392 392 392 156 156 156 156 61 61 61 61 25 25 25 25. 0.0. 1753 1753 1753 1753 1108 1108 1108 1108 862 862 862 862 626 626 626 626 392 392 392 392 156 156 156 156 61 61 61 61 25 25 25 25. 0.5. 1753 1753 1753 1753 1108 1108 1108 1108 862 862 862 862 626 626 626 626 392 392 392 392 156 156 156 156 61 61 61 61 25 25 25 25. 1.0. CNp. 1753 1753 1753 1753 1108 1108 1108 1108 862 862 862 862 626 626 626 626 392 392 392 392 156 156 156 156 61 61 61 61 25 25 25 25. 1.5 1753 1753 1753 1753 1108 1108 1108 1108 862 862 862 862 626 626 626 626 392 392 392 392 156 156 156 156 61 61 61 61 25 25 25 25. 2.0 1397 1516 1575 1611 909 975 1008 1028 716 765 789 804 527 560 576 586 331 351 361 368 127 137 142 145 46 51 54 55 16 19 20 21. 0.0 1720 1733 1739 1742 1104 1106 1106 1107 861 861 862 862 626 626 626 626 392 392 392 392 157 157 156 156 61 61 61 61 25 25 25 25. 0.5 1753 1753 1753 1753 1109 1108 1108 1108 862 862 862 862 626 626 626 626 392 392 392 392 157 157 156 156 61 61 61 61 25 25 25 25. 1.0. CNpk. 1754 1753 1753 1753 1109 1108 1108 1108 863 862 862 862 627 626 626 626 392 392 392 392 157 157 157 156 61 61 61 61 25 25 25 25. 1.5. 0.0 1332 1332 1332 1332 969 969 969 969 789 789 789 789 594 594 594 594 381 381 381 381 154 154 154 154 61 61 61 61 24 24 24 24. 2.0 ∗∗∗ 1754 1753 1753 ∗∗∗ 1109 1108 1108 ∗∗∗ 862 862 862 ∗∗∗ 626 626 626 ∗∗∗ 392 392 392 ∗∗∗ 157 157 156 ∗∗∗ 61 61 61 ∗∗∗ 25 25 25 985 985 985 985 714 714 714 714 588 588 588 588 452 452 452 452 299 299 299 299 126 126 126 126 50 50 50 50 20 20 20 20. 0.5 524 524 524 524 392 392 392 392 331 331 331 331 264 264 264 264 183 183 183 183 81 81 81 81 33 33 33 33 14 14 14 14. 1.0. CNpm. 286 286 286 286 222 222 222 222 192 192 192 192 156 156 156 156 111 111 111 111 51 51 51 51 21 21 21 21 09 09 09 09. 1.5 174 174 174 174 139 139 139 139 121 121 121 121 100 100 100 100 72 72 72 72 33 33 33 33 14 14 14 14 06 06 06 06. 2.0. Table 12(b) Relative bias (%) of CNp CNpk CNpm , and CNpmk for t distribution, t7. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 1067 1156 1200 1225 794 852 881 899 655 700 722 736 499 531 547 556 322 342 351 357 151 135 140 143 46 51 53 55 16 19 20 21. 0.0 1065 1033 1019 1012 755 739 732 728 613 603 599 596 465 460 457 456 304 302 301 301 127 127 126 126 51 51 51 50 21 21 21 21. 0.5 659 592 569 558 449 420 411 406 366 348 343 304 282 273 270 268 190 187 185 185 83 82 82 82 34 34 34 33 14 14 14 14. 1.0. CNpmk. 464 345 321 311 303 249 239 234 243 209 202 199 185 166 162 160 124 116 114 113 55 52 52 51 23 22 21 21 09 09 09 09. 1.5. ∗∗∗ 237 205 195 ∗∗∗ 169 154 149 ∗∗∗ 141 131 127 ∗∗∗ 111 105 104 ∗∗∗ 77 75 74 ∗∗∗ 35 34 34 ∗∗∗ 15 14 14 ∗∗∗ 06 06 06. 2.0. Accuracy Analysis of the Percentile Method 677.

(27) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 21 21 21 21. 0.0. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 21 21 21 21. 0.5. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 21 21 21 21. 1.0. CNp. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 21 21 21 21. 1.5 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 21 21 21 21. 2.0 1422 14687 1489 1501 913 927 949 938 716 723 726 728 525 527 528 528 333 333 333 333 139 139 139 139 54 54 54 54 21 21 21 21. 0.0 1448 1484 1501 1501 917 930 948 950 716 723 726 728 523 525 527 527 331 331 332 332 138 139 139 139 54 54 54 54 21 21 21 21. 0.5 1532 1539 1541 1543 942 947 947 949 727 730 731 732 525 527 528 528 330 331 332 332 138 139 139 139 54 54 54 54 21 21 21 21. 1.0. CNpk. 1522 1536 1540 1542 936 945 701 949 722 729 731 732 522 526 528 528 329 331 331 332 138 139 139 139 54 54 54 54 22 21 21 21. 1.5 1475 1532 1539 1541 902 942 750 701 699 727 730 731 508 525 527 528 321 330 331 332 138 138 139 139 54 54 54 54 22 21 21 21. 2.0 947 947 947 947 701 701 701 701 577 577 577 577 441 441 441 441 290 290 290 290 125 125 125 125 49 49 49 49 20 20 20 20. 0.0 1080 1080 1080 1080 750 750 750 750 604 604 604 604 454 454 454 454 295 295 295 295 127 127 127 127 50 50 50 50 20 20 20 20. 0.5 556 556 556 556 399 399 399 399 330 330 330 330 257 257 257 257 175 175 175 175 79 79 79 79 32 32 32 32 13 13 13 13. 1.0. CNpm. 249 249 249 249 197 197 197 197 169 169 169 169 136 136 136 136 96 96 96 96 45 45 45 45 19 19 19 19 08 08 08 08. 1.5 133 133 133 133 110 110 110 110 96 96 96 96 79 79 79 79 57 57 57 57 27 27 27 27 11 11 11 11 05 05 05 05. 2.0. Table 13(a) CNpk CNpm , and CNpmk for chi-square distribution, 2 3 Relative bias (%) of CNp . Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 912 925 931 934 691 695 696 697 576 577 577 577 445 443 442 442 293 292 291 291 126 126 126 126 50 50 50 50 20 20 20 20. 0.0 1060 1067 1070 1073 744 747 747 748 603 603 604 604 455 454 454 454 296 296 295 295 127 127 127 127 50 50 50 50 20 20 20 20. 0.5 632 598 5835 579 431 417 412 409 350 341 338 336 267 263 261 260 179 177 177 176 80 80 80 80 32 32 32 32 13 13 13 13. 1.0. CNpmk. 299 270 263 259 223 208 204 202 186 176 173 172 146 140 139 138 101 98 97 97 46 45 45 45 19 19 19 19 08 08 08 08. 1.5. 185 144 139 137 143 117 114 113 120 101 99 98 93 82 81 80 63 58 58 57 29 27 27 27 12 11 11 11 05 05 05 05. 2.0. 678 Wu et al..

(28) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 1304 1304 1304 1304 807 807 807 807 623 623 623 623 451 451 451 451 284 284 284 284 119 119 119 119 46 46 46 46 18 18 18 18. 0.0. 1304 1304 1304 1304 807 807 807 807 623 623 623 623 451 451 451 451 284 284 284 284 119 119 119 119 46 46 46 46 18 18 18 18. 0.5. 1304 1304 1304 1304 807 807 807 807 623 623 623 623 451 451 451 451 284 284 284 284 119 119 119 119 46 46 46 46 18 18 18 18. 1.0. CNp. 1304 1304 1304 1304 807 807 807 807 623 623 623 623 451 451 451 451 284 284 284 284 119 119 119 119 46 46 46 46 18 18 18 18. 1.5 1304 1304 1304 1304 807 807 807 807 623 623 623 623 451 451 451 451 284 284 284 284 119 119 119 119 46 46 46 46 18 18 18 18. 2.0 1135 1195 1224 1240 737 762 774 780 584 598 604 608 434 440 443 444 280 281 282 282 119 119 119 119 46 46 46 46 18 18 18 18. 0.0 1225 1254 1268 1275 782 791 795 798 612 616 618 619 447 448 449 449 283 283 283 283 119 119 119 119 46 46 46 46 18 18 18 18. 0.5 1294 1299 1300 1301 800 803 804 805 619 621 621 622 448 449 450 450 282 283 283 283 119 119 119 119 46 46 46 46 18 18 18 18. 1.0. CNpk. 1288 1297 1300 1301 796 802 804 805 615 620 621 622 446 449 449 450 281 283 283 283 119 119 119 119 46 46 46 46 18 18 18 18. 1.5 1246 1295 1299 1300 797 800 803 804 595 619 621 621 433 448 449 450 274 282 283 283 117 119 119 119 46 46 46 46 19 18 18 18. 2.0 844 844 844 844 622 622 622 622 511 511 511 511 390 390 390 390 256 256 256 256 111 111 111 111 44 44 44 44 17 17 17 17. 0.0 856 856 856 856 598 598 598 598 483 483 483 483 364 364 364 364 238 238 238 238 104 104 104 104 41 41 41 41 16 16 16 16. 0.5 470 470 470 470 328 328 328 328 269 269 269 269 207 207 207 207 140 140 140 140 63 63 63 63 25 25 25 25 10 10 10 10. 1.0. CNpm. 225 225 225 225 169 169 169 169 142 142 142 142 113 113 113 113 78 78 78 78 36 36 36 36 15 15 15 15 06 06 06 06. 1.5 124 124 124 124 97 97 97 97 83 83 83 83 67 67 67 67 47 47 47 47 22 22 22 22 09 09 09 09 04 04 04 04. 2.0. Table 13(b) Relative bias (%) of CNp CNpk CNpm , and CNpmk for chi-square distribution, 2 6. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 757 787 803 811 578 594 601 605 486 495 499 502 380 384 385 386 255 256 256 256 112 111 111 111 44 44 44 44 17 17 17 17. 0.0 861 859 858 858 607 604 602 601 492 489 487 486 371 368 367 367 241 240 240 239 104 104 104 104 41 41 41 41 16 16 16 16. 0.5 570 524 507 498 371 352 344 341 295 283 279 276 221 215 213 211 146 143 142 142 65 64 64 64 26 26 26 26 10 10 10 10. 1.0. CNpmk. 315 262 248 241 214 187 180 177 171 154 150 148 129 119 113 116 86 81 80 80 39 37 37 37 16 15 15 15 06 06 06 06. 1.5. 309 153 140 135 197 113 106 103 151 94 89 87 107 73 70 69 66 50 49 48 29 23 23 23 12 10 09 09 05 04 04 04. 2.0. Accuracy Analysis of the Percentile Method 679.

(29) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 2785 2785 2785 2785 1743 1743 1743 1743 1350 1350 1350 1350 976 976 976 976 610 610 610 610 247 247 247 247 98 98 98 98 40 40 40 40. 0.0. 2785 2785 2785 2785 1743 1743 1743 1743 1350 1350 1350 1350 976 976 976 976 610 610 610 610 247 247 247 247 98 98 98 98 40 40 40 40. 0.5. 2785 2785 2785 2785 1743 1743 1743 1743 1350 1350 1350 1350 976 976 976 976 610 610 610 610 247 247 247 247 98 98 98 98 40 40 40 40. 1.0. CNp. 2785 2785 2785 2785 1743 1743 1743 1743 1350 1350 1350 1350 976 976 976 976 610 610 610 610 247 247 247 247 98 98 98 98 40 40 40 40. 1.5 2785 2785 2785 2785 1743 1743 1743 1743 1350 1350 1350 1350 976 976 976 976 610 610 610 610 247 247 247 247 98 98 98 98 40 40 40 40. 2.0 2677 2716 2735 2745 1717 1726 1731 1734 1342 1344 1346 1346 975 975 976 976 611 611 611 611 247 247 247 247 98 98 98 98 40 40 40 40. 0.0 2719 2743 2754 2760 1725 1732 1735 1737 1342 1345 1346 1347 973 974 975 975 609 610 610 610 246 246 246 246 98 98 98 98 40 40 40 40. 0.5 2776 2780 2781 2782 1736 1739 1741 1741 1345 1347 1348 1348 973 974 975 975 609 609 610 610 246 246 246 246 98 98 98 98 40 40 40 40. 1.0. CNpk. 2770 2778 2781 2782 1732 1738 1740 1741 1342 1346 1348 1348 971 974 975 975 608 609 610 610 246 246 246 246 98 98 98 98 40 40 40 40. 1.5 2740 2776 2780 2781 1710 1736 1739 1741 1327 1345 1347 1348 962 973 974 975 604 609 610 610 245 246 246 246 97 98 98 98 40 40 40 40. 2.0 1783 1783 1783 1783 1325 1325 1325 1325 1092 1092 1092 1092 834 834 834 834 546 546 546 546 228 228 228 228 92 92 92 92 37 37 37 37. 0.0 2023 2023 2023 2023 1414 1414 1414 1414 1141 1141 1141 1141 858 858 858 858 555 555 555 555 231 231 231 231 92 92 92 92 38 38 38 38. 0.5 872 872 872 872 692 692 692 692 597 597 597 597 484 484 484 484 340 340 340 340 153 153 153 153 64 64 64 64 27 27 27 27. 1.0. CNpm. 407 407 407 407 350 350 350 350 313 313 313 313 264 264 264 264 194 194 194 194 90 90 90 90 39 39 39 39 16 16 16 16. 1.5 228 228 228 228 202 202 202 202 183 183 183 183 157 157 157 157 117 117 117 117 55 55 55 55 24 24 24 24 10 10 10 10. 2.0. Table 14(a) CNpk CNpm , and CNpmk for F distribution, F10 10 Relative bias (%) of CNp . Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 1778 1780 1781 1781 1332 1330 1328 1328 1101 1098 1096 1095 841 838 837 836 549 548 547 547 229 228 228 228 92 92 92 92 37 37 37 37. 0.0 2042 2035 2032 2030 1425 1421 1419 1418 1149 1146 1145 1144 862 860 860 859 557 556 556 556 231 231 231 231 93 93 93 93 38 38 38 38. 0.5 922 900 891 887 714 704 701 699 610 604 602 601 490 487 486 486 342 341 341 341 154 153 153 153 64 64 64 64 27 27 27 27. 1.0. CNpmk. 428 416 413 411 361 355 353 352 320 316 315 314 268 265 265 264 195 194 194 194 91 90 90 90 39 39 39 39 17 17 17 17. 1.5. 223 227 227 227 205 203 202 202 187 184 184 184 159 158 157 157 118 117 117 117 55 55 55 55 24 24 24 24 10 10 10 10. 2.0. 680 Wu et al..

(30) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 1499 1499 1499 1499 936 936 936 936 725 725 725 725 526 526 526 526 332 332 332 332 139 139 139 139 54 54 54 54 22 22 22 22. 0.0. 1499 1499 1499 1499 936 936 936 936 725 725 725 725 526 526 526 526 332 332 332 332 139 139 139 139 54 54 54 54 22 22 22 22. 0.5. 1499 1499 1499 1499 936 936 936 936 725 725 725 725 526 526 526 526 332 332 332 332 139 139 139 139 54 54 54 54 22 22 22 22. 1.0. CNp. 1499 1499 1499 1499 936 936 936 936 725 725 725 725 526 526 526 526 332 332 332 332 139 139 139 139 54 54 54 54 22 22 22 22. 1.5 1499 1499 1499 1499 936 936 936 936 725 725 725 725 526 526 526 526 332 332 332 332 139 139 139 139 54 54 54 54 22 22 22 22. 2.0 1329 1389 1418 1435 868 892 903 910 688 701 707 711 511 516 519 521 329 330 331 331 139 139 139 139 54 54 54 54 22 22 22 22. 0.0 1423 1451 1464 1471 913 922 925 928 715 719 721 722 523 524 525 525 331 332 332 332 139 139 139 139 54 54 54 54 22 22 22 22. 0.5 1490 1494 1495 1496 930 932 933 934 721 723 724 724 524 525 526 526 331 331 332 332 138 139 139 139 54 54 54 54 22 22 22 22. 1.0. CNpk. 1484 1493 1495 1496 926 932 933 934 718 722 723 724 522 525 525 526 330 331 332 332 138 139 139 139 54 54 54 54 22 22 22 22. 1.5 1447 1490 1494 1495 899 930 932 933 700 721 723 724 511 524 525 526 324 331 331 332 137 138 139 139 54 54 54 54 22 22 22 22. 2.0 1000 1000 1000 1000 738 738 738 738 607 607 607 607 463 463 463 463 304 304 304 304 130 130 130 130 52 52 52 52 21 21 21 21. 0.0 1002 1002 1002 1002 705 705 705 705 571 571 571 571 432 432 432 432 283 283 283 283 122 122 122 122 49 49 49 49 19 19 19 19. 0.5 530 530 530 530 381 381 381 381 316 316 316 316 247 247 247 247 168 168 168 168 76 76 76 76 31 31 31 31 73 73 73 73. 1.0. CNpm. 254 254 254 254 197 197 197 197 168 168 168 168 135 135 135 135 95 95 95 95 44 44 44 44 18 18 18 18 07 07 07 07. 1.5 141 141 141 141 114 114 114 114 99 99 99 99 81 81 81 81 58 58 58 58 27 27 27 27 11 11 11 11 05 05 05 05. 2.0. Table 14(b) Relative bias (%) of CNp CNpk CNpm , and CNpmk for F distribution, F10 40. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 914 945 959 968 696 711 718 722 584 592 596 598 454 457 459 460 303 303 304 304 131 131 131 131 52 52 52 52 21 21 21 21. 0.0 1012 1008 1006 1005 716 712 710 709 581 577 576 575 439 436 435 434 286 285 284 284 123 123 123 123 49 49 49 49 20 20 19 19. 0.5 627 583 566 558 422 403 396 393 341 329 325 323 260 254 252 250 174 171 170 170 78 77 77 77 31 31 31 31 13 13 13 13. 1.0. CNpmk. 338 289 276 270 238 214 207 204 195 179 175 173 150 141 139 138 102 98 97 96 46 45 45 45 19 19 18 18 08 08 08 08. 1.5. 308 168 156 151 204 128 122 119 159 108 104 102 116 86 84 83 74 60 59 59 32 28 28 28 13 12 11 11 05 05 05 05. 2.0. Accuracy Analysis of the Percentile Method 681.

(31) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 612 612 612 612 348 348 348 348 257 257 257 257 178 178 178 178 107 107 107 107 44 44 44 44 16 16 16 16 06 06 06 06. 0.0. 612 612 612 612 348 348 348 348 257 257 257 257 178 178 178 178 107 107 107 107 44 44 44 44 16 16 16 16 06 06 06 06. 0.5. 612 612 612 612 348 348 348 348 257 257 257 257 178 178 178 178 107 107 107 107 44 44 44 44 16 16 16 16 06 06 06 06. 1.0. CNp. 612 612 612 612 348 348 348 348 257 257 257 257 178 178 178 178 107 107 107 107 44 44 44 44 16 16 16 16 06 06 06 06. 1.5 612 612 612 612 348 348 348 348 257 257 257 257 178 178 178 178 107 107 107 107 44 44 44 44 16 16 16 16 06 06 06 06. 2.0 553 573 583 589 315 326 331 335 233 241 245 248 162 167 170 171 98 101 102 103 42 42 43 43 16 16 16 16 06 06 06 06. 0.0 611 611 611 611 347 347 348 348 257 257 257 257 177 177 177 178 106 106 107 107 44 44 44 44 16 16 16 16 06 06 06 06. 0.5 610 611 611 611 347 347 348 348 257 257 257 257 177 177 177 177 106 106 106 107 44 44 44 44 16 16 16 16 06 06 06 06. 1.0. CNpk. 609 611 611 611 346 347 347 348 256 257 257 257 177 177 177 177 106 106 106 107 44 44 44 44 16 16 16 16 06 06 06 06. 1.5 525 610 611 611 287 347 347 348 212 257 257 257 148 177 177 177 93 106 106 106 38 44 44 44 12 16 16 16 03 06 06 06. 2.0 302 302 302 302 212 212 212 212 172 172 172 172 130 130 130 130 84 84 84 84 37 37 37 37 14 14 14 14 05 05 05 05. 0.0 9.0 9.0 9.0 9.0 5.6 5.6 5.6 5.6 4.2 4.2 4.2 4.2 2.9 2.9 2.9 2.9 1.8 1.8 1.8 1.8 0.8 0.8 0.8 0.8 0.3 0.3 0.3 0.3 0.1 0.1 0.1 0.1. 0.5 2.1 2.1 2.1 2.1 1.4 1.4 1.4 1.4 1.1 1.1 1.1 1.1 0.8 0.8 0.8 0.8 0.5 0.5 0.5 0.5 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0. 1.0. CNpm. 0.9 0.9 0.9 0.9 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0. 1.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0. 2.0. Table 15(a) CNpk CNpm , and CNpmk for beta distribution, B3 20 Relative bias (%) of CNp . Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 264 277 283 287 186 195 199 202 152 158 162 164 116 120 123 124 76 79 80 81 35 36 36 37 14 14 14 14 05 05 05 05. 0.0 106 100 97 95 63 60 59 58 47 45 44 44 33 31 31 31 20 19 19 19 08 08 08 08 03 03 03 03 01 01 01 01. 0.5 3.0 2.6 2.4 2.3 1.9 1.6 1.6 1.5 1.4 1.3 1.2 1.2 1.0 1.0 0.9 0.8 0.6 0.5 0.5 0.5 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0. 1.0. CNpmk. 1.7 1.2 1.0 1.0 1.1 0.8 0.7 0.7 0.8 0.6 0.5 0.5 0.6 0.4 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0. 1.5. 133 07 06 05 82 05 04 04 58 04 03 03 37 02 02 02 21 02 01 01 06 01 01 01 −01 00 00 00 −02 00 00 00. 2.0. 682 Wu et al..

(32) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 686 686 686 686 399 399 399 399 298 298 298 298 208 208 208 208 127 127 127 127 53 53 53 53 20 20 20 20 07 07 07 07. 0.0. 686 686 686 686 399 399 399 399 298 298 298 298 208 208 208 208 127 127 127 127 53 53 53 53 20 20 20 20 07 07 07 07. 0.5. 686 686 686 686 399 399 399 399 298 298 298 298 208 208 208 208 127 127 127 127 53 53 53 53 20 20 20 20 07 07 07 07. 1.0. CNp. 686 686 686 686 399 399 399 399 298 298 298 298 208 208 208 208 127 127 127 127 53 53 53 53 20 20 20 20 07 07 07 07. 1.5 686 686 686 686 399 399 399 399 298 298 298 298 208 208 208 208 127 127 127 127 53 53 53 53 20 20 20 20 07 07 07 07. 2.0 429 515 558 584 245 297 322 348 183 221 241 252 127 154 168 176 76 93 101 106 30 37 41 44 11 14 15 16 05 06 06 06. 0.0 636 656 664 669 386 391 393 395 294 295 296 297 207 208 208 208 126 126 126 126 53 53 53 53 20 20 20 20 08 08 07 07. 0.5 683 685 685 685 398 398 398 398 297 298 298 298 208 208 208 208 126 126 126 126 53 53 53 53 20 20 20 20 08 08 07 07. 1.0. CNpk. 684 685 686 686 397 398 398 398 297 298 298 298 207 208 208 208 126 126 126 126 53 53 53 53 20 20 20 20 08 08 08 07. 1.5 633 685 686 686 365 398 398 398 275 297 298 298 193 208 208 208 113 126 126 126 54 53 53 53 27 20 20 20 15 08 08 07. 2.0 374 374 374 374 265 265 265 265 216 216 216 216 163 163 163 163 106 106 106 106 47 47 47 47 18 18 18 18 07 07 07 07. 0.0 364 364 364 364 240 240 240 240 187 187 187 187 136 136 136 136 86 86 86 86 37 37 37 37 14 14 14 14 05 05 05 05. 0.5 253 253 253 253 153 153 153 153 116 116 116 116 82 82 82 82 51 51 51 51 22 22 22 22 08 08 08 08 03 03 03 03. 1.0. CNpm. 143 143 143 143 87 87 87 87 66 66 66 66 47 47 47 47 29 29 29 29 13 13 13 13 05 05 05 05 02 02 02 02. 1.5 8.4 8.4 8.4 8.4 5.3 5.3 5.3 5.3 4.1 4.1 4.1 4.1 2.9 2.9 2.9 2.9 1.8 1.8 1.8 1.8 0.8 0.8 0.8 0.8 0.3 0.3 0.3 0.3 0.1 0.1 0.1 0.1. 2.0 197 256 286 303 139 182 203 215 115 149 166 176 88 113 126 133 58 74 82 87 24 32 36 38 09 12 14 14 04 05 06 06. 0.0. Table 15(b) Relative bias (%) of CNp CNpk CNpm , and CNpmk for beta distribution, B3 35. Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 4069 391 383 379 274 260 254 251 214 203 199 196 153 147 144 142 94 91 89 89 40 39 38 38 15 15 14 14 06 06 06 06. 0.5 410 333 306 293 227 191 178 172 163 140 132 128 109 96 91 89 63 57 55 54 26 24 23 23 10 09 09 09 04 03 03 03. 1.0. CNpmk. 369 220 190 176 196 125 110 103 137 91 81 77 89 61 56 53 49 36 33 32 19 15 14 14 07 06 05 05 03 02 02 02. 1.5. 3807 168 126 112 1929 95 74 67 1292 69 55 50 775 46 38 35 383 26 22 21 131 11 09 09 48 04 04 03 21 02 01 01. 2.0. Accuracy Analysis of the Percentile Method 683.

(33) 50. 30. 20. 10. 3000. 1000. 300. 100. n. 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5. d/. − T/. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 22 22 22 22. 0.0. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 22 22 22 22. 0.5. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 22 22 22 22. 1.0. CNp. 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 22 22 22 22. 1.5 1547 1547 1547 1547 953 953 953 953 734 734 734 734 530 530 530 530 333 333 333 333 139 139 139 139 54 54 54 54 22 22 22 22. 2.0 1422 1468 1489 1501 913 927 934 938 716 723 726 728 525 527 528 528 333 333 333 333 139 139 139 139 54 54 54 54 22 22 22 22. 0.0 1448 1484 1501 1510 917 930 936 940 716 723 726 728 523 525 527 527 331 331 332 332 139 139 139 139 54 54 54 54 22 22 22 22. 0.5 1532 1539 1541 1543 942 947 949 950 727 730 731 732 525 527 528 528 330 331 3325 332 138 139 139 139 54 54 54 54 22 22 22 22. 1.0. CNpk. 1522 1536 1540 1542 936 945 948 949 727 729 731 732 522 527 528 528 329 331 3325 332 138 139 139 139 54 54 54 54 22 22 22 22. 1.5 1475 1532 1539 1541 903 942 947 949 699 727 730 731 508 525 527 528 321 330 331 332 136 138 139 139 54 54 54 54 22 22 22 22. 2.0 947 947 947 947 701 701 701 701 577 577 577 577 441 441 441 441 290 290 290 290 125 125 125 125 50 50 50 50 20 20 20 20. 0.0 1080 1080 1080 1080 750 750 750 750 604 604 604 604 454 454 454 454 295 295 295 295 127 127 127 127 50 50 50 50 20 20 20 20. 0.5 556 556 556 556 399 399 399 399 330 330 330 330 257 257 257 257 175 175 175 175 80 80 80 80 32 32 32 32 13 13 13 13. 1.0. CNpm. 249 249 249 249 197 197 197 197 169 169 169 169 136 136 136 136 96 96 96 96 45 45 45 45 19 19 19 19 08 08 08 08. 1.5 133 133 133 133 110 110 110 110 96 96 96 96 79 79 79 79 57 57 57 57 27 27 27 27 11 11 11 11 05 05 05 05. 2.0. Table 16(a) CNpk CNpm , and CNpmk for gamma distribution, G15 1 Relative bias (%) of CNp . Downloaded by [National Chiao Tung University ] at 00:39 26 April 2014. 912 925 931 934 691 695 696 697 576 577 577 577 445 443 442 442 294 292 291 291 126 126 126 126 50 50 50 50 20 20 20 20. 0.0 1060 1067 1071 1073 744 747 748 748 603 603 604 604 455 454 454 454 296 296 295 295 127 127 127 127 50 50 50 50 20 20 20 20. 0.5 632 598 585 579 431 417 412 409 350 341 338 336 267 263 261 260 179 177 177 176 80 80 80 80 33 33 33 33 13 13 13 13. 1.0. CNpmk. 299 270 263 259 223 208 204 202 186 176 173 172 146 140 139 138 100 98 97 97 46 46 45 45 19 19 19 19 08 08 08 08. 1.5. 185 144 139 137 143 117 114 113 120 101 99 98 93 82 81 80 63 58 58 57 29 28 27 27 12 11 11 11 05 05 05 05. 2.0. 684 Wu et al..